The University of British Columbia
Final Examination - December 5, 2008
Mathematics 257/316
All Sections
Closed book examination
Last Name:
(USE CAPITALS)
Student Number
Time: 2.5 hours
, First:
Signature
Instructor’s Name & Section
Special Instructions:
- Students are not allowed to bring any notes into the exam.
- No calculators are allowed.
Rules governing examinations
• Each candidate must be prepared to produce, upon request, a
UBCcard for identification.
• Candidates are not permitted to ask questions of the invigilators,
except in cases of supposed errors or ambiguities in examination
questions.
• No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
• Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action.
(a) Having at the place of writing any books, papers
or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness
shall not be received.
• Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any
examination material from the examination room without permission of the invigilator.
• Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
Page 1 of 15 pages
1
20
2
25
3
15
4
20
5
20
Total
100
December 5, 2008
Math 257/316
Name:
Page 2 of 15 pages
[20] 1. Consider the differential equation
4x2 y 00 + xy 0 − (1 + 3x)y = 0
(1)
(a) Classify the points 0 ≤ x < ∞ as ordinary points, regular singular points, or
irregular singular points.
∞
P
(b) Find two values of r such that there are solutions of the form y(x) =
n=0
an xn+r .
(c) Use the series expansion in (b) to determine two independent solutions of (1).
You only need to calculate the first three non-zero terms in each case.
(d) Determine the radius of convergence of the series in (c).
December 5, 2008
Math 257/316
(Question 1 Continued)
Name:
Page 3 of 15 pages
December 5, 2008
Math 257/316
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Page 4 of 15 pages
[25] 2. Consider the following initial boundary value problem for the heat equation:
ut = uxx , 0 < x < 1, t > 0
ux (0, t) = 1 and u(1, t) = 0
u(x, 0) = cos(3πx/2), 0 < x < 1
(2)
(a) Determine a steady state solution to the boundary value problem. [5 marks]
(b) Use this steady state solution to determine the solution to the boundary value
problem (2) by separation of variables. [10 marks]
(c) Briefly describe how you would use the method of finite differences to obtain
an approximate solution this boundary value problem.
Use the notation
ukn ' u(xn , tk ) to represent the nodal values on the finite difference mesh.
Explain how you propose to approximate the boundary conditions. [10 marks]
¡¡
¢ ¢
R1
4
πx dx = π2 (2n+1)
Hint: It might be useful to know that 0 (1 − x) cos 2n+1
2
2
December 5, 2008
Math 257/316
(Question 2 Continued)
Name:
Page 5 of 15 pages
December 5, 2008
Math 257/316
(Question 2 Continued)
Name:
Page 6 of 15 pages
December 5, 2008
Math 257/316
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Page 7 of 15 pages
[15] 3. The displacement u(x, t) of a string of length 1 subject to viscous damping satisfies
the damped wave equation
utt + 2ut = uxx
The string is set in motion from its initial displacement u = f (x) from rest while both the
ends of the string are held fixed. Use separation of variables to solve for the displacement of
the string as a function of time by solving the following boundary value problem:
utt + 2ut = uxx
0 < x < 1, t > 0
u(0, t) = 0,
u(1, t) = 0
u(x, 0) = f (x),
ut (x, 0) = 0
Since f (x) is not specified, you may leave the expressions for the Fourier coefficients as
integrals.
December 5, 2008
Math 257/316
(Question 3 Continued)
Name:
Page 8 of 15 pages
December 5, 2008
Math 257/316
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Page 9 of 15 pages
[20] 4. Use separation of variables to solve the following mixed boundary value problem
for the semi-circular region:
1
1
urr + ur + 2 uθθ = 0,
r
r
0 < r < a,
0<θ<π
∂u
(r, π) = 0
∂θ
u(r, θ) < ∞ as r → 0 and u(a, θ) = 1
u(r, 0) = 0
and
December 5, 2008
Math 257/316
(Question 4 Continued)
Name:
Page 10 of 15 pages
December 5, 2008
Math 257/316
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Page 11 of 15 pages
[20] 5. Solve the following inhomogeneous initial boundary value problem for the heat
equation:
ut = uxx + e−2t sin (5πx) + (1 − x), 0 < x < 1, t > 0
u(0, t) = t and u(1, t) = 0
u(x, 0) = sin (πx)
December 5, 2008
Math 257/316
(Question 5 Continued)
Name:
Page 12 of 15 pages
December 5, 2008
(Additional Page)
Math 257/316
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December 5, 2008
(Additional Page)
Math 257/316
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December 5, 2008
(Additional Page)
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